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HARMONIC UNIVALENT FUNCTIONS INVOLVING

FOX - WRIGHT

SAIBAH SIREGAR, NORHEZAN UMAR ANDTN. AZMAR TN.DAUD

Abstract.In this paper we introduce the new subclass of analytic and harmonic univalent func-tions involving Fox-Wright funcfunc-tions.The coefficient bounds and Growth Theorem as well as Dis-tortion Theorem results for these functions are obtained.

1. INTRODUCTION

A continuous functionf(u,v) =u+ivis a complex-valued harmonic function

in a simply connected complex domain Cif both u and v are real harmonic in

C. LetH denote the family of function f =h+gwhich are harmonic and sense

-preserving in the open unit diskU=

z:|z|<1 wherehandgare given by

h(z) =z+ ∞

k=2

akzk , g(z) = ∞

k=1

bkzk (1)

The functionhis called the analytic part and g is called the co-analytic part

of the harmonic function f=h+g. The classHreduces to the class of normalized

univalent analytic functions, if the co-analytic part of f is zero. A necessary and

sufficient condition for f inH to be locally univalent and sense-preserving inU

is thath′(z) >

g

′(z)

inU . LetT H be the class of functions inH that may be

expressed as f=h+gwhere Received 06-12-2016, Accepted 26-12-2016. 2010 Mathematics Subject Classification: 30C45

Key words and Phrases: Harmonic, Univalent, starlike, Fox-Wright functions.

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h(z) =z− ∞

k=2

akzk , g(z) = ∞

k=1

bkzk ,ak≥0,bk≥0 (2)

LetSH(α)andKH(α)be the subclasses ofHconsisting of univalent harmonic

functions starlike of order and convex of orderα, respectively, where0≤α<1

and|z|=r<1, if ∂

∂ θ(arg f(re

))α

, |z|=r<1, (3)

and

∂ ∂ θ

" arg

∂ ∂ θ f(re

)

#

≥α, |z|=r<1, (4)

Also letT SH(α)andT KH(α) be the respective subclasses ofSH(α)andKH(α) consisting of functions of the form (2).

Harmonic functions are indeed famous for their use in the study of minimal sur-faces and play an important role in a variety of problems in applied mathematics.

Harmonic functions have been studied by different geometers such as Kneser [8].

In 1984, Clunie and Sheil-Small [2] began a study of complex-valued, harmonic mappings defined on a domainU⊂C.

The coefficients bounds for the classesSH(α),KH(α),T SH(α),T KH(α)a are

stud-ied in details by Silverman [11], Jahangiri [4], Jahangiri & Silverman [5], Silverman

& Silvia [12],Jakubowski et al [6], Janteng et. al [7], Yalcin, [13], [14], [15] and Siregar et.al [10].

2. PRELIMINARY RESULTS In this paper, we considered the function as follows

Imα,k(α1,β1)f(z) =z+ ∞

k=2

Ωm

kΘmkakzk, (5)

Ωmk = 

     

    

q

j=1

Γ αj+Aj(k−1)

s

j=1

Γ βj+Bj(k−1)

    

1 (k−1)!

     

, (6)

Θmk = "

(k−1)k(k+λ2)! λ!(k−2)!

#

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In this section, the classesSH(Ω,Θ,γ)andTH(Ω,Θ,γ), of functions which

are harmonic univalent inUwill be introduced. Some properties for functions f

belonging to these classes which include the coefficient estimates, growth results and distortion theorem will be given.

Let f =h+gdenote in the form (2). f be in class function ofSH(Ω,Θ,γ)if satisfy the condition

Re

3. COEFFICIENT BOUNDS Theorem 1.Let f =h+g with h and g given by (2),

By using triangle inequality for the above function is,

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using the properties of sigma, from equation (11), we obtain

equation (6) and (7) respectively.

f(z1)−f(z2)

≥ |z1−z2|(1−|b1| −|z2||γ| −|b1|)>0

Consequently, f is univalent inU. To prove that f is sense preserving inU. This is

because

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NextImλ,k(α1,β1)f(z)in equation (5),

The harmonic mappings

f(z) =z+

The restriction placed in Theorem 1 on the moduli of the coefficients of f=h+g

enables us to conclude for arbitrary rotation of the coefficients of fthat the resulting

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Next the condition (10) is also necessary for functions f to be inTH(Ω,Θ,γ). Theorem 2.Let f =h+gwithhandggiven by (2). Then f TH(Ω,Θ,γ)if and

only if the inequality (9) holds for the coefficients of f =h+g.

Proof. First suppose that f ∈TH(Ω,Θ,γ), then by (8) have Re

n

(Imλ,kh(z))′−Imλ,kg(z)o

=Re (

1− ∞

k=2

k[Ωm

kΘmk]akzk−1− ∞

k=1

k[Ωm

kΘmk]bkzk−1 )

>1−|γ|

If choosezto be real and letz→1−, then we can have1− ∞

k=2

k[Ωm

kΘmk]|ak| − ∞

k=1

k[Ωm

kΘmk]|bk|>1−|γ|, which is precisely the assertion (9). Conversely, suppose that the inequality (9) holds true. Then can be find from the equation (8) that

Re n

(Imλ,kh(z))′−Imλ,kg(z)o

=Re (

1− ∞

k=2

k[ΩmkΘmk]akzk−1− ∞

k=1

k[ΩmkΘmk]bkzk−1 )

≥2− ∞

k=1

k[ΩmkΘmk](|ak|+|bk|)|z|k−1

>2− ∞

k=1

k[ΩmkΘmk](|ak|+|bk|)≥1−|γ| provided that the inequality (9) is satisfied.

4. GROWTH BOUNDS AND DISTORTION THEOREM

In this subsection, growth bounds for functions inTH(Ω,Θ,γ)will be obtained and extreme points for this class will be given.

Theorem 3.If f TH(Ω,Θ,γ)for0<|γ| ≤1,N0,λ≤0and|z|=r>1, then

f(z)

≤(1+|b1|)r+

|γ| −|b1| 2[Ωm

kΘmk] r2

and

f(z)

≥(1−|b1|)r−

|γ| −|b1| 2[Ωm

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Proof. Let f∈TH(Ω,Θ,γ). Taking the absolute value of f(z). The right hand side

f(z)

≥ (1+|b1|)r− ∞

k=2

[|ak|+|bk|]rk

≥ (1+|b1|)r− ∞

k=2

[|ak|+|bk|]r2

≥(1+|b1|)r−|γ| −|b1| 2[Ωm

kΘmk] ∞

k=2

2[Ωm kΘmk]

|γ| −|b1|[|ak|+|bk|]r 2

By equation (9), we obtain

f(z)

≥(1+|b1|)r−

|γ| −|b1| 2[Ωm

kΘmk]

r2 (14)

The left hand side,

f(z)

≤ (1+|b1|)r+ ∞

k=2

[|ak|+|bk|]rk

≤ (1+|b1|)r+ ∞

k=2

[|ak|+|bk|]r2

≤(1+|b1|)r+

|γ| −|b1| 2[Ωm

kΘmk] r2

k=2

2[Ωm kΘmk]

|γ| −|b1|[|ak|+|bk|]r 2

And also, by equation (9), we find

f(z)

≤(1+|b1|)r+

|γ| −|b1| 2[Ωm

kΘmk]

r2 (15)

The proof is complete.

Distortion for function inTH(Ω,Θ,γ)will be obtained by Theorem 4.

Theorem 4.If f TH(Ω,Θ,γ)for0<|γ| ≤1, m∈N0,λ0and|z|=r>1

f′(z)

≤(1+|b1|) +

|γ| −|b1|

Ωm 2Θm2

r,

and

f′(z)

≥(1−|b1|)−

|γ| −|b1|

Ωm 2Θm2

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To obtained distortion theorem, it can be differentiate the equation in (14) and

(15), then

f′(z)

≤(1+|b1|) +

|γ| −|b1|

Ωm 2Θm2

r,

and

f′(z)

≥(1−|b1|)−

|γ| −|b1|

Ωm 2Θm2

r.

REFERENCES

1. Al-Saqsi, K., Darus, M. (2008). An Operator Defined by Convolution Involving

the Polylogarithms Functions. University Kebangsaan Malaysia.

2. Clunie, J. & Sheil-Small, T. 1984. Harmonic univalent functions. Ann. Acad. Sci. Fenn.,Ser. A.I9: 3-25.

3. Darus, M. & Siregar, S. 2005. Certain subclass of harmonic functions using

Hadamard product. Proc. Int. AdvancedTech. Congress. ITMA(Univ. Putra

Malaysia).

4. Jahangiri, J. M. 1999. Harmonic functions starlike in the unite disk. J. math.

Anal. Appl.235: 470-477.

5. Jahangiri, J. M. and Silverman, H. 2002. Harmonic Univalent functions with

Varying Arguments,Int.J. Appl. Math. 8(3): 267-275.

6. Jakubowski, Z. J., Majchrzak, W. & Skalska, K. 1993. Harmonic mappings with

a positive real part.Materialy Konferencjiz Teorii Zagadnien Ekstremal- nych,

LodzXIV: 17-24.

7. Janteng, A., Halim, S. A. & Darus, M. 2007. A new subclass of harmonic

univalent functions.South. Asian Bull. Math. 31: 81-88.

8. Kneser, H. 1926. L”osung der aufgabe 41. Jahresber, Deutsch. Math.-Verein.

35:123-124.

9. Ponnusamy, S. & Sabapathy, S. 1996. Polylogarithms in the theory of univalent functions.Results in Mathematics30: 136-150.

10. Siregar, S, Darus, M and Jahangiri, J.M. 2007. Harmonic Univalent functions

Defined by Convolution,Int.J. Comp. & Math. Appl. 1(2): 173-182.

11. Silverman, H. 1998. Harmonic univalent functions with negative coefficients.J. Math. Anal. Appl. 220(1): 275-284.

12. Silverman, H. & Silvia, E. M. 1999. Subclasses of harmonic univalent functions. New Zeal. J. Math. 28: 275-284.

13. Yalcin, S. & O”ztu”rk, M. 2006. On a subclass of certain convex harmonic

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14. Yalcin, S., O”ztu”rk, M. & Yamankaradeniz, M. 2000. On some subclasses of

harmonic functions. Kluwer Acad. Publ., Math. Appl., Fun. Equ. Ineq. 518:

325-331.

15. Yalcin, S., O”ztu”rk, M. & Yamankaradeniz, M. 2002. A new subclass of

harmonic mappings with positive real part.Hacettepe J. Math. Stat.31: 13-18.

Saibah Siregar: Department of Science and Biotechnology, Faculty of Engineering and

Life Sciences, University of Selangor, Bestari Jaya 45600, Selangor D.E. Malaysia

E-mail: saibah@unisel.edu.my

Norhezan Umar: Department of Science and Biotechnology, Faculty of Engineering and

Life Sciences, University of Selangor, Bestari Jaya 45600, Selangor D.E. Malaysia

E-mail: norhezanumar@unisel.edu.my

Tn. Azmar Tn.Daud: Department of Science and Biotechnology, Faculty of Engineering

and Life Sciences, University of Selangor, Bestari Jaya 45600, Selangor D.E. Malaysia

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